A comparison theorem for state-dependent regime-switching diffusion processes is established, which enables us to pathwise-control the evolution of the state-dependent switching component simply by Markov chains. Moreover, a sharp estimate on the stability of Markovian regime-switching processes under the perturbation of transition rate matrices is provided. Our approach is based on elaborate constructions of switching processes in the spirit of Skorokhod’s representation theorem varying according to the problem being dealt with. In particular, this method can cope with switching processes in an infinite state space and not necessarily of birth–death type. As an application, some known results on the ergodicity and stability of state-dependent regime-switching processes can be improved.

In this work, the secular evolution of exoplanetary systems is investigated, when the variability of the masses of celestial bodies is the leading factor of dynamical evolution. The masses of the parent star and the planets change due to the particles leaving the bodies and falling on them. At the same time, bodies masses are assumed to change isotropically at different rates. The law of mass change is considered to be known and given function of time. The relative motions of the planets are investigated by the methods of the canonical perturbation theory in the absence of resonances. It is assumed that the orbits of the planets do not intersect. Evolutionary equations in analogues of Poincaré variables (Λi, λi, ξi, ηi, pi, qi) are obtained and used to study the K2-3 exoplanetary system. All analytical and numerical calculations are performed with the aid of the Wolfram Mathematica.

We studied the problem of two spherical celestial bodies in the general case when the masses of the bodies change non-isotropically at different rates in the presence of reactive forces. The problem was investigated by methods of perturbation theory based on aperiodic motion along a quasi-conic section, using the equation of perturbed motion in the form of Newton’s equations. The problem is described by the variables a, e, i, π, ω, λ, which are analogs of the corresponding Keplerian elements and the equations of motion in these variables are obtained. Averaging over the mean longitude, we obtained the evolution equations of the two-body problem with variable masses in the presence of reactive forces. The obtained evolution equations have the exact analytic integral ${a^3 e^4 = a^3_0 e^4_0} = {const}$.

We introduce a unified framework for solving first passage times of time-homogeneous diffusion processes. Using potential theory and perturbation theory, we are able to deduce closed-form truncated probability densities, as asymptotics or approximations to the original first passage time densities, for single-side level crossing problems. The framework is applicable to diffusion processes with continuous drift functions; in particular, for bounded drift functions, we show that the perturbation series converges. In the present paper, we demonstrate examples of applying our framework to the Ornstein–Uhlenbeck, Bessel, exponential-Shiryaev, and hypergeometric diffusion processes (the latter two being previously studied by Dassios and Li (2018) and Borodin (2009), respectively). The purpose of this paper is to provide a fast and accurate approach to estimating first passage time densities of various diffusion processes.

We consider a linear operator pencil with complex parameter mapping one Hilbert space onto another. It is known that the resolvent is analytic in an open annular region of the complex plane centred at the origin if and only if the coefficients of the Laurent series satisfy a doubly-infinite set of left and right fundamental equations and are suitably bounded. If the resolvent has an isolated singularity at the origin we propose a recursive orthogonal decomposition of the domain and range spaces that enables us to construct the key nonorthogonal projections that separate the singular and regular components of the resolvent and subsequently allows us to find a formula for the basic solution to the fundamental equations. We show that each Laurent series coefficient in the singular part of the resolvent can be approximated by a weakly convergent sequence of finite-dimensional matrix operators and we show how our analysis can be extended to find a global expression for the resolvent of a linear pencil in the case where the resolvent has only a finite number of isolated singularities.

We analyse a class of chemical reaction networks under mass-action kinetics involving multiple time scales, whose deterministic and stochastic models display qualitative differences. The networks are inspired by gene-regulatory networks and consist of a slow subnetwork, describing conversions among the different gene states, and fast subnetworks, describing biochemical interactions involving the gene products. We show that the long-term dynamics of such networks can consist of a unique attractor at the deterministic level (unistability), while the long-term probability distribution at the stochastic level may display multiple maxima (multimodality). The dynamical differences stem from a phenomenon we call noise-induced mixing, whereby the probability distribution of the gene products is a linear combination of the probability distributions of the fast subnetworks which are ‘mixed’ by the slow subnetworks. The results are applied in the context of systems biology, where noise-induced mixing is shown to play a biochemically important role, producing phenomena such as stochastic multimodality and oscillations.

In this paper, the concept of perturbation theory is applied to derive a general electric field (E-field) expression for any arbitrary-shaped microstrip patch antenna. The arbitrary shape is created by adding small perturbation in a regular patch shape, which is used to find perturbed and unperturbed electromagnetic wave solutions for resultant E-field of patch antenna. Ansoft HFSS simulator is used to validate the derived field expression in curvilinear coordinates for a regular circular-shaped patch. Then the proposed field analysis is applied to develop two new arbitrary-shaped patches in C-band for desired E-field patterns.

We investigate the effects of toxins on the multiple coexistence solutions of an unstirred chemostat model of competition between plasmid-bearing and plasmid-free organisms when the plasmid-bearing organism produces toxins. It turns out that coexistence solutions to this model are governed by two limiting systems. Based on the analysis of uniqueness and stability of positive solutions to two limiting systems, the exact multiplicity and stability of coexistence solutions of this model are established by means of the combination of the fixed-point index theory, bifurcation theory and perturbation theory.

We give sufficient conditions for the following problem: given a topological space $X$, a metric space $Y$, a subspace $Z$ of $Y$, and a continuous map $f$ from $X$ to $Y$, is it possible, by applying to $f$ an arbitrarily small perturbation, to ensure that $f\left( {{X}^{'}} \right)$ does not meet $Z$? We also give a relative variant: if $f\left( X\prime \right)$ does not meet $Z$ for a certain subset ${X}'\subset X$, then we may keep $f$ unchanged on ${X}'$. We also develop a variant for continuous sections of fibrations and discuss some applications to matrix perturbation theory.

Following the derivation of amplitude equations through a new two-time-scale method [O'Malley, R. E., Jr. & Kirkinis, E (2010) A combined renormalization group-multiple scale method for singularly perturbed problems. Stud. Appl. Math. 124, 383–410], we show that a multi-scale method may often be preferable for solving singularly perturbed problems than the method of matched asymptotic expansions. We illustrate this approach with 10 singularly perturbed ordinary and partial differential equations.

In this work, we review the analytical and semi-analytical tools introduced to deal with resonant proper elements and their applications to the Trojan asteroids, the numerical computation of synthetic proper elements for resonant and non resonant asteroids, and the introduction of proper elements for planet crossing asteroids. We discuss the applications and accuracy of these methods and present some comparisons between them.To search for other articles by the author(s) go to: http://adsabs.harvard.edu/abstract_service.html

We study irreducible time-homogenous Markov chains with finite state space in discrete time. We obtain results on the sensitivity of the stationary distribution and other statistical quantities with respect to perturbations of the transition matrix. We define a new closeness relation between transition matrices, and use graph-theoretic techniques, in contrast with the matrix analysis techniques previously used.

The gating mechanism of a single ion channel is usually modelled by a continuous-time Markov chain with a finite state space, partitioned into two classes termed ‘open’ and ‘closed’. It is possible to observe only which class the process is in. A burst of channel openings is defined to be a succession of open sojourns separated by closed sojourns all having duration less than t0. Let N(t) be the number of bursts commencing in (0, t]. Then are measures of the degree of temporal clustering of bursts. We develop two methods for determining the above measures. The first method uses an embedded Markov renewal process and remains valid when the underlying channel process is semi-Markov and/or brief sojourns in either the open or closed classes of state are undetected. The second method uses a ‘backward’ differential-difference equation.

The observed channel process when brief sojourns are undetected can be modelled by an embedded Markov renewal process, whose kernel is shown, by exploiting connections with bursts when all sojourns are detected, to satisfy a differential-difference equation. This permits a unified derivation of both exact and approximate expressions for the kernel, and leads to a thorough asymptotic analysis of the kernel as the length of undetected sojourns tends to zero.